The $Q_{ \alpha}$-restriction problem

Let $\alpha \in [0, 1)$ and $\Omega$ be an open connected subset of $\mathbb{R}^{n \geq 2}$. This paper shows that the $Q_{\alpha}$-restriction problem $Q_{\alpha} \vert {}_{\Omega} = \mathscr{Q}_{\alpha} (\Omega)$ is solvable if and only if $\Omega$ is an Ahlfors $n$-regular domain; i.e., $\operatorname{vol} \bigl ( B(x, r) \cap \Omega \bigr ) \gtrsim r^n$ for any Euclidean ball $B(x, r)$ with center $x \in \Omega$ and radius $r \in \bigl ( 0, \operatorname{diam} (\Omega) \bigr ) $ , thereby not only yielding an exponential $Q_{\alpha}$-integrability as a proper adjustment of the John–Nirenberg type inequality for $Q_{\alpha}$ conjectured in [3, Problem 8.1, (8.2)] but also resolving the quasiconformal extension problem for $Q_{\alpha}$ posed in [3, Problem 8.5].

Keywords

$Q$ space, restriction, Ahlfors regular domain, uniform domain, Minkowski type dimension

2010 Mathematics Subject Classification

42B35, 46E35

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Z. Wang and Y. Zhou were supported by the National Natural Science Foundation of China (#11871088). J. Xiao was supported by the NSERC of Canada.

Received 24 May 2018

Accepted 5 July 2018

Published 30 April 2020